Over the past few years, there has been a significant amount of research focused on studying the ReLU activation function, with the aim of achieving neural network convergence through over-parametrization. However, recent developments in the field of Large Language Models (LLMs) have sparked interest in the use of exponential activation functions, specifically in the attention mechanism. Mathematically, we define the neural function $F: \mathbb{R}^{d \times m} \times \mathbb{R}^d \rightarrow \mathbb{R}$ using an exponential activation function. Given a set of data points with labels $\{(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)\} \subset \mathbb{R}^d \times \mathbb{R}$ where $n$ denotes the number of the data. Here $F(W(t),x)$ can be expressed as $F(W(t),x) := \sum_{r=1}^m a_r \exp(\langle w_r, x \rangle)$, where $m$ represents the number of neurons, and $w_r(t)$ are weights at time $t$. It's standard in literature that $a_r$ are the fixed weights and it's never changed during the training. We initialize the weights $W(0) \in \mathbb{R}^{d \times m}$ with random Gaussian distributions, such that $w_r(0) \sim \mathcal{N}(0, I_d)$ and initialize $a_r$ from random sign distribution for each $r \in [m]$. Using the gradient descent algorithm, we can find a weight $W(T)$ such that $\| F(W(T), X) - y \|_2 \leq \epsilon$ holds with probability $1-\delta$, where $\epsilon \in (0,0.1)$ and $m = \Omega(n^{2+o(1)}\log(n/\delta))$. To optimize the over-parameterization bound $m$, we employ several tight analysis techniques from previous studies [Song and Yang arXiv 2019, Munteanu, Omlor, Song and Woodruff ICML 2022].
相關內容
This paper considers the specification of covariance structures with tail estimates. We focus on two aspects: (i) the estimation of the VaR-CoVaR risk matrix in the case of larger number of time series observations than assets in a portfolio using quantile predictive regression models without assuming the presence of nonstationary regressors and; (ii) the construction of a novel variable selection algorithm, so-called, Feature Ordering by Centrality Exclusion (FOCE), which is based on an assumption-lean regression framework, has no tuning parameters and is proved to be consistent under general sparsity assumptions. We illustrate the usefulness of our proposed methodology with numerical studies of real and simulated datasets when modelling systemic risk in a network.
The importance of unspanned macroeconomic variables for Dynamic Term Structure Models has been intensively discussed in the literature. To our best knowledge the earlier studies considered only linear interactions between the economy and the real-world dynamics of interest rates in DTSMs. We propose a generalized modelling setup for Gaussian DTSMs which allows for unspanned nonlinear associations between the two and we exploit it in forecasting. Specifically, we construct a custom sequential Monte Carlo estimation and forecasting scheme where we introduce Gaussian Process priors to model nonlinearities. Sequential scheme we propose can also be used with dynamic portfolio optimization to assess the potential of generated economic value to investors. The methodology is presented using US Treasury data and selected macroeconomic indices. Namely, we look at core inflation and real economic activity. We contrast the results obtained from the nonlinear model with those stemming from an application of a linear model. Unlike for real economic activity, in case of core inflation we find that, compared to linear models, application of nonlinear models leads to statistically significant gains in economic value across considered maturities.
We study the 2D Navier-Stokes equation with transport noise subject to periodic boundary conditions. Our main result is an error estimate for the time-discretisation showing a convergence rate of order (up to) 1/2. It holds with respect to mean square error convergence, whereas previously such a rate for the stochastic Navier-Stokes equations was only known with respect to convergence in probability. Our result is based on uniform-in-probability estimates for the continuous as well as the time-discrete solution exploiting the particular structure of the noise.
The solution of computational fluid dynamics problems is one of the most computationally hard tasks, especially in the case of complex geometries and turbulent flow regimes. We propose to use Tensor Train (TT) methods, which possess logarithmic complexity in problem size and have great similarities with quantum algorithms in the structure of data representation. We develop the Tensor train Finite Element Method -- TetraFEM -- and the explicit numerical scheme for the solution of the incompressible Navier-Stokes equation via Tensor Trains. We test this approach on the simulation of liquids mixing in a T-shape mixer, which, to our knowledge, was done for the first time using tensor methods in such non-trivial geometries. As expected, we achieve exponential compression in memory of all FEM matrices and demonstrate an exponential speed-up compared to the conventional FEM implementation on dense meshes. In addition, we discuss the possibility of extending this method to a quantum computer to solve more complex problems. This paper is based on work we conducted for Evonik Industries AG.
The recent popularity of large language models (LLMs) has brought a significant impact to boundless fields, particularly through their open-ended ecosystem such as the APIs, open-sourced models, and plugins. However, with their widespread deployment, there is a general lack of research that thoroughly discusses and analyzes the potential risks concealed. In that case, we intend to conduct a preliminary but pioneering study covering the robustness, consistency, and credibility of LLMs systems. With most of the related literature in the era of LLM uncharted, we propose an automated workflow that copes with an upscaled number of queries/responses. Overall, we conduct over a million queries to the mainstream LLMs including ChatGPT, LLaMA, and OPT. Core to our workflow consists of a data primitive, followed by an automated interpreter that evaluates these LLMs under different adversarial metrical systems. As a result, we draw several, and perhaps unfortunate, conclusions that are quite uncommon from this trendy community. Briefly, they are: (i)-the minor but inevitable error occurrence in the user-generated query input may, by chance, cause the LLM to respond unexpectedly; (ii)-LLMs possess poor consistency when processing semantically similar query input. In addition, as a side finding, we find that ChatGPT is still capable to yield the correct answer even when the input is polluted at an extreme level. While this phenomenon demonstrates the powerful memorization of the LLMs, it raises serious concerns about using such data for LLM-involved evaluation in academic development. To deal with it, we propose a novel index associated with a dataset that roughly decides the feasibility of using such data for LLM-involved evaluation. Extensive empirical studies are tagged to support the aforementioned claims.
BCART (Bayesian Classification and Regression Trees) and BART (Bayesian Additive Regression Trees) are popular Bayesian regression models widely applicable in modern regression problems. Their popularity is intimately tied to the ability to flexibly model complex responses depending on high-dimensional inputs while simultaneously being able to quantify uncertainties. This ability to quantify uncertainties is key, as it allows researchers to perform appropriate inferential analyses in settings that have generally been too difficult to handle using the Bayesian approach. However, surprisingly little work has been done to evaluate the sensitivity of these modern regression models to violations of modeling assumptions. In particular, we will consider influential observations, which one reasonably would imagine to be common -- or at least a concern -- in the big-data setting. In this paper, we consider both the problem of detecting influential observations and adjusting predictions to not be unduly affected by such potentially problematic data. We consider three detection diagnostics for Bayesian tree models, one an analogue of Cook's distance and the others taking the form of a divergence measure and a conditional predictive density metric, and then propose an importance sampling algorithm to re-weight previously sampled posterior draws so as to remove the effects of influential data in a computationally efficient manner. Finally, our methods are demonstrated on real-world data where blind application of the models can lead to poor predictions and inference.
Most machine learning methods require tuning of hyper-parameters. For kernel ridge regression with the Gaussian kernel, the hyper-parameter is the bandwidth. The bandwidth specifies the length-scale of the kernel and has to be carefully selected in order to obtain a model with good generalization. The default methods for bandwidth selection is cross-validation and marginal likelihood maximization, which often yields good results, albeit at high computational costs. Furthermore, the estimates provided by these methods tend to have very high variance, especially when training data are scarce. Inspired by Jacobian regularization, we formulate an approximate expression for how the derivatives of the functions inferred by kernel ridge regression with the Gaussian kernel depend on the kernel bandwidth. We then use this expression to propose a closed-form, computationally feather-light, bandwidth selection heuristic based on controlling the Jacobian. In addition, the Jacobian expression illuminates how the bandwidth selection is a trade-off between the smoothness of the inferred function, and the conditioning of the training data kernel matrix. We show on real and synthetic data that compared to cross-validation and marginal likelihood maximization, our method is considerably faster and considerably more stable in terms of bandwidth selection.
This paper deals with inference in a class of stable but nearly-unstable processes. Autoregressive processes are considered, in which the bridge between stability and instability is expressed by a time-varying companion matrix $A_{n}$ with spectral radius $\rho(A_{n}) < 1$ satisfying $\rho(A_{n}) \rightarrow 1$. This framework is particularly suitable to understand unit root issues by focusing on the inner boundary of the unit circle. Consistency is established for the empirical covariance and the OLS estimation together with asymptotic normality under appropriate hypotheses when $A$, the limit of $A_n$, has a real spectrum, and a particular case is deduced when $A$ also contains complex eigenvalues. The asymptotic process is integrated with either one unit root (located at 1 or $-1$), or even two unit roots located at 1 and $-1$. Finally, a set of simulations illustrate the asymptotic behavior of the OLS. The results are essentially proved by $L^2$ computations and the limit theory of triangular arrays of martingales.
Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.
Pre-trained deep neural network language models such as ELMo, GPT, BERT and XLNet have recently achieved state-of-the-art performance on a variety of language understanding tasks. However, their size makes them impractical for a number of scenarios, especially on mobile and edge devices. In particular, the input word embedding matrix accounts for a significant proportion of the model's memory footprint, due to the large input vocabulary and embedding dimensions. Knowledge distillation techniques have had success at compressing large neural network models, but they are ineffective at yielding student models with vocabularies different from the original teacher models. We introduce a novel knowledge distillation technique for training a student model with a significantly smaller vocabulary as well as lower embedding and hidden state dimensions. Specifically, we employ a dual-training mechanism that trains the teacher and student models simultaneously to obtain optimal word embeddings for the student vocabulary. We combine this approach with learning shared projection matrices that transfer layer-wise knowledge from the teacher model to the student model. Our method is able to compress the BERT_BASE model by more than 60x, with only a minor drop in downstream task metrics, resulting in a language model with a footprint of under 7MB. Experimental results also demonstrate higher compression efficiency and accuracy when compared with other state-of-the-art compression techniques.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191